
There are canonical isomorphisms \(\Psi _{\overline{\mathcal {J}}_{g,n}}:\Sigma (\overline{\mathcal {J}}_{g,n})\xrightarrow{} J^{trop}_{g,n}\)  and \(\overline{\Psi }_{\overline{\mathcal {J}}_{g,n}}:\overline{\Sigma }(\overline{\mathcal {J}}_{g,n})\xrightarrow{} \overline{J}^{trop}_{g,n}\)  of, respectively, generalized cone complexes and generalized extended cone complexes, in such a way that the map (that we baptise the modular analytic tropicalization map)
\(\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}}=\overline{\Psi }_{\overline{\mathcal {J}}_{g,n}}\circ \operatorname{trop}_{\overline{\mathcal {J}}_{g,n}}\colon \big |\overline{\mathcal {J}}_{g,n}^{\beth }\big |\xrightarrow{} \overline{\Sigma }(\overline{\mathcal {J}}_{g,n}) \xrightarrow[\cong ]{\overline{\Psi }_{\overline{\mathcal {J}}_{g,n}}} \overline{J}^{trop}_{g,n}\) 
has a modular description (see Definition REF ).
Moreover, the diagram (REF ) commutes, via the suitable forgetful-stabilization morphisms, with the analogous diagram for \(\overline{\mathcal {M}}_{g,n}\)  established in [1]}.

The two maps
\({\big |\overline{\mathcal {J}}_{g,n}\big | && \big |\overline{\mathcal {J}}_{g,n}^{\beth }\big | [ll]_{\operatorname{red}_{\overline{\mathcal {J}}_{g,n}}} [rr]^{\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}}} && \overline{J}^{trop}_{g,n}}\) 
are compatible with the toroidal stratification (REF ) of \(\overline{\mathcal {J}}_{g,n}\)  and the stratification (REF ) of \(\overline{J}^{trop}_{g,n}\)  as a generalized extended cone complex.
Moreover, the diagram (REF ) is compatible with the restrictions to \(\overline{\mathcal {J}}_{g,n, d}\) , \(\overline{\mathcal {J}}_{g,n}^{\operatorname{spl}}\)  and \(\overline{\mathcal {J}}_{g,n}(\phi )\)  for any universal stability condition \(\phi \in V_{g,n}\) .


